1. Field of the Invention
The present invention relates to a method of determining a calibration curve and an analysis method and apparatus using this calibration curve. Particularly, the present invention relates to a method of determining a regression function of a calibration curve for quantitative analysis of an analyte, with good accuracy in all the concentration region, when the calibration curve having a sigmoidal shape is used.
2. Description of the Related Art
In the conventional quantitative analysis of an analyte contained in a liquid sample, the concentration of the analyte in a test sample is determined from an analytical measured value (absorbancy, transmitting or reflected optical density, other physical measuring quantity, or a signal indicating such a physical measuring quantity), after it is subjected to a proper chemical or enzymatic reaction. In such a method, it is common practice to determine the concentration of an analyte by using a calibration curve (also commonly referred to as standard curve or working curve) which has been preliminarily drawn by plotting the interrelation between the known concentrations of the analyte in the standard samples and the analytical measured values such as optical densities of the standard samples. When the calibration curve has an adequate linearity over a wider range in the region of quantitative analysis, the calibration curve can be prepared with a relatively smaller number of standard samples, which are near the upper limit, lower limit and in the intermediate point in the determination range of the quantitative analysis.
In practice, however, there are many calibration curves which are not linear in general. Examples of such calibration curves include those in immunological reactions such as enzyme immunoassay (EIA). In the immunoassay, the antigen-antibody reaction, which is the basis of the measurement, is essentially reversible in accordance with the law of mass action; hence, the calibration curve tends to be S-shaped sigmoid. This sigmoid type of calibration curve is also found in an enzyme reaction system in which the binding constant between the enzyme- and the substrate varies due to formation of the enzyme-substrate complex, and in an allosteric enzyme reaction system involving regulatory function such as end-product inhibition. Furthermore, in the enzyme immunoassay, the shape of calibration curve varies easily depending on the type of the measurement system and the reaction condition. In addition, because the enzyme immunoassay is the ultramicro analysis method for micro substance, measured data have relatively broad dispersion. In the enzyme immunoassay, therefore, a large number of standard samples covering the entire determination range in the quantitative analysis have to be prepared and analyzed to draw a calibration curve.
In the field of clinical examinations necessitating quickness, economy and simplicity, it has been desired to prepare a calibration curve of high accuracy with a possible minimum number of standard samples. Hence, tries to prepare a calibration curve have been made up to now by getting a regression formula, from which the calibration curve is prepared with measurement values of a smaller number of standard samples.
A regression model of a calibration curve may be basically classified into two types: theoretical formula and empirical formula. Since a theoretical formula is rarely applicable to practical cases and has difficulty in statistical handling usually due to a complicated non-linear function of higher degree, an empirical formula based on an actual measurement is often used. Empirical formulae may also be classified basically into two types. One is based on a non-split-plot experiment, in which a single regression function covers a whole calibration curve. The other is based on a split-plot experiment, in which the calibration curve is segmented and a number of regression equations are calculated for each segmented portion.
Split-plot experiments include: a linear interpolation method in which interpolation is made on the basis of linear segments obtained by connecting two adjacent points on a calibration curve by a straight line; and a fitting method by spline function in which all intervals between two adjacent points are covered by cubic polynominal functions, maintaining continuity with adjoining functions. Both methods as well as the other approaches cannot fit S-shaped curve of calibration satisfactorily without many analytical measured values.
As non-split-plot experiments, there are known a regression method using a logistic curve; a regression method using logit-log conversion formula; a method using an equilateral hyperbola; and a method applying a polynomial expression of cubic or higher degree to a sigmoid calibration curve. Among them, the method using an equilateral hyperbola cannot be fitted well to an S-shaped calibration curve.
Logistic curves have been known as an empirical formula for S-shaped curves. A most prevalent logistic curve is represented by following formula having four coefficients: ##EQU1## wherein, x: concentration,
y: analytical measured value (data such as optical density), and PA1 a, b, c, and d : coefficients. PA1 wherein said calibration curve is split into at least three parts as followings: PA1 and wherein the adjacent two parts of the calibration curve have an identical slope in the boundaries of respective concentration regions of the above. PA1 (a) representing the calibration curve for a low concentration region by a multi-degree function; PA1 (b) representing the calibration curve for an intermediate concentration region by an exponential function; PA1 (c) representing the calibration curve for a high concentration region by a multi-degree function; and PA1 (d) assuming boundary conditions of respective concentration regions of the above that the adjacent two parts of the calibration curve have an identical slope at the boundary points, whereby the functions of the calibration curve for respective concentration regions are determined. PA1 (a) plotting respective concentrations (p.sub.i) of the analyte in the standard sample and the logarithmic values (log q.sub.i) of the corresponding analytical measured values (q.sub.i) on the rectangular coordinate system; PA1 (b) finding an intermediate concentration region (p.sub.1 -p.sub.2) which constitutes a linear portion in this semi-logarithmic graph; PA1 (c) expressing the part of calibration curve of this intermediate concentration region (P.sub.1 -p.sub.2) as EQU Y=exp (b.multidot.X+d) PA1 which means EQU X=(ln Y-d)/b (1) PA1 wherein Y is the analytical measured value (q.sub.i) such as optical density or other physical quantity, PA1 (d) expressing the part of calibration curve of the low concentration region (from the minimum concentration p.sub.0 in the standard samples to the concentration p.sub.1) as EQU X=e.multidot.Y.sup.2 +f.multidot.Y+g (2) PA1 wherein e, f and g are coefficients; PA1 (e) expressing the part of calibration curve of the high concentration region (from the concentration P.sub.2 to the maximum concentration p.sub.3 in the standard samples) as EQU X=l.multidot.Y.sup.2 +m.multidot.Y+n (3) PA1 wherein l, m and n are coefficients; PA1 (f) setting boundary conditions where the differentiated value (dX/dY) of Equation (1) at the coordinate (p.sub.1, q.sub.1) is equal to the differentiated value (dX/dY) of Equation (2) at the coordinate (p.sub.1, q.sub.1) and where the differentiated value (dX/dY) of Equation (2) at the coordinate (p.sub.2, q.sub.2) is equal to the differentiated value (dX/dY) of Equation (3) at the coordinate (p.sub.2, q.sub.2); and PA1 (g) calculating respective coefficients in Equations (1), (2) and (3) and finding therefrom the continuous regression function of the calibration curve for the whole concentration region covering from p.sub.0 to P.sub.3. PA1 (a) providing an intermediate concentration region (p.sub.1 -p.sub.2) which is preliminarily defined as the region of linear portion in a semi-logarithmic graph in which respective concentrations (p.sub.i) of the analyte in the standard samples and the logarithmic values (log q.sub.i) of the corresponding analytical measured values (q.sub.i) on a rectangular coordinate system, said linear portion having the most proximate concentrations p.sub.1 and p.sub.2 at both ends; PA1 (b) expressing the part of calibration curve of an intermediate concentration region (p.sub.1 -p.sub.2) as EQU Y=exp (b.multidot.X+d) PA1 which means EQU X=(ln Y-d)/b (1) PA1 wherein Y is the analytical measured value (q.sub.i) such as optical density or other physical measuring quantity, PA1 (c) expressing the part of calibration curve of the low concentration region (from the minimum concentration p.sub.0 in the standard sample to the concentration p.sub.1) as EQU X=e.multidot.Y.sup.2 +f.multidot.Y+g (2) PA1 wherein e, f and g are coefficients; PA1 (d) expressing the part of calibration curve of the high concentration region (from the concentration p.sub.2 to the maximum concentration p.sub.3 in the standard sample) as EQU X=l.multidot.Y.sup.2 +m.multidot.Y+n (3) PA1 wherein l, m and n are coefficients; PA1 (e) setting boundary conditions where the differentiated value (dX/dY) of Equation (1) at the coordinate (p.sub.1, q.sub.1) is equal to the differentiated value (dX/dY) of Equation (2) at the coordinate (p.sub.1, q.sub.1) and where the differentiated value (dX/dY) of Equation (2) at the coordinate (p.sub.2, q.sub.2) is equal to the differentiated value (dx/dY) of Equation (3) at the coordinate (p.sub.2, q.sub.2); and PA1 (f) calculating respective coefficients in Equations (1), (2) and (3) and finding therefrom the continuous regression function of the calibration curve for the whole concentration region covering from p.sub.0 to p.sub.3. PA1 1) input means for inputting the known concentrations (p.sub.i) of the analytes contained in plural standard samples and their analytical measured values (q.sub.i); PA1 2) a processor for receiving the data input from said input means and processing the operations mentioned below to determine a regression function of the calibration curve on the basis of the input (p.sub.i) and (q.sub.i), said operations including steps of; PA1 3) a calibration curve generator for generating the calibration curve to be used for quantitative analysis of the analyte contained in a test sample, the calibration curve being prepared from the regression function determined by said processor.
(Rodbard et al.: Statistical analysis of radioimmunoassays and immunoradiometric (labeled antibody) assays. A generalized, weighted, iterative, least-squares method for logistic curve fitting. Symposium on RIA and Related Procedures in Medicine, p165, Int. Atomic Energy Agency, Vienna, 1974)
This logistic curve is a sigmoid curve as shown in FIG. 1 and is excellent as a calibration curve model obtained from a small number of measurement points and standard samples, since this curve gives not only a linear part in its middle but also curve parts at both ends and furthermore an asymptotic part outside the end. Statistical treatment of this logistic curve is complicated, however, because the above formula which represents this curve is nonlinear and the regression of this formula requires an iterative least square method for. In addition, it is necessary to obtain the analytical measured value (signal, or .DELTA.OD in the Example hereinafter) precisely when the amount of the antigen (concentration: x) is zero and infinity (.infin.) respectively. In order to obtain the measurement point for infinite amount of the antigen, it is required to prepare and store a standard sample containing a large excessive amount of the antigen. As a matter of fact, this is extremely difficult as compared to preparation and storage of standard samples containing a normal amount of the antigen (analyte). Furthermore, it is likely that the signal changes due to hook effect (also referred to as prozone effect) and the calibration curve has a maximum value (or minimum value) when an excessive amount of the antigen exists against antibody at an immunoassay. In such a case, the signal for infinite amount of the antigen cannot be obtained.
A logit-log conversion is to apply a logit conversion for the vertical axis (indicated or measured value) of a calibration graph and apply a logarithm conversion to the horizontal axis (concentration: dose) of the calibration graph. Thus, a S-shaped curve can be linearized by use of the following simple linear polynominal: ##EQU2## wherein, B.sub.0 and B.sub.x are the measured values at concentration 0 (zero) and concentration x respectively; and N is the measured value at infinite concentration.
(Rodbard et al.: Rapid calculation of radioimmunoassay results, J. Lab. Clin. Med., 74, p770, 1969)
The logit-log conversion can be performed in the simpler regression by a simple least square method and is superior to logistic curve described above. Even this logit-log conversion, however, has a defect in that more deviation from the linear line will occur, unless both signal B.sub.0 at zero concentration and signal N at infinite concentration are determined accurately. In other words, it is difficult to prepare a calibration curve when the number of measurement points is small.
A most prevalent conventional method is approximation of a sigmoid calibration curve by a cubic (three-degree) polynomial. This method is effective in case of using only a linear part of an S-shaped curve as the working range of a calibration curve. However, this method cannot fit the whole range of data points including not only the linear part in the S-shaped curve but also curve parts at both ends and furthermore an asymptotic part outside the end. Thus, an effort to improve accuracy in the middle of the sigmoid curve results in lowering accuracy at both ends. On the contrary, an effort to apply both ends of the curve as working ranges results in sacrificing accuracy to a certain extent at the middle part of the curve. These are not improved even when more measurement points of standard samples are used. There has been a certain limit to a method of applying one model (function) to a whole calibration curve.